Teaching Continuum Mechanics in a
Mechanical Engineering Program
Yucheng Liu
University of Louisiana at Lafayette
1. Introduction
Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuum. Modeling an
object as a continuum assumes that the substance of the object completely fills the space
it occupies, which is a simplifying assumption
for analysis purposes. As emphasized by
Mase [1], continuum mechanics is the fundamental basis upon which several graduate engineering courses are founded. These courses
include elasticity, plasticity, viscoelasticity, and
fluid mechanics. Gollub [2] also demonstrated
the important role that continuum mechanics
plays in contemporary physics. Therefore, it is
beneficial to teach the principles of continuum
mechanics to first-year graduate students or
upper-level undergraduates to provide them
with the necessary background in continuum
theory so that they can readily pursue a formal
course in any of the aforementioned subjects.
Due to its importance in engineering education, continuum mechanics has been built
into the engineering curriculum and taught at
many prominent universities, such as MIT,
Texas A&M University, Carnegie Mellon, etc.
Lagoudas et al. [3] demonstrated the significance of teaching continuum mechanics
in their engineering program and designed a
core continuum mechanics course for sophomore students at Texas A&M. In that course,
conservation laws and fundamental concepts
of continuum mechanics were taught using
computer tools [4]. This paper describes the
design of the course, Continuum Mechanics,
for the Mechanical Engineering (ME) program
at the University of Louisiana at Lafayette (UL
Lafayette). Compared to similar courses taught
in other universities, this course emphasizes
the application of mathematics in formulating
and solving fundamental equations of fluid and
solid mechanics, as detailed in the course syllabus.
The paper is organized as follows. Section 2 describes the objectives of this course
and briefly explains how the course objectives
correlate with the ME program objectives.
Section 3 presents the detailed methodology
and approach of teaching this course, which
include topics (concepts, principles, laws, and
equations), class organization, and evaluation
instruments. Section 4 discusses the outcomes
of teaching this course at the University of Louisville and indicates the modification of course
structure since then. Finally, this paper is concluded and ended with section 5.
2. Course Objectives
Continuum Mechanics is a three-credit
course which emphasizes the mathematics
and analysis methods used in the study of the
behavior of a continuous medium. The course
is designed to be taken by first-year graduate
students of the ME department at UL Lafayette.
The primary objectives of this course are:
1. To study the conservation principles in the mechanics of continua and formulate the equations that describe the motion and mechanical
behaviors of continuum materials, and
2. To present the applications of these equations to simple problems associated with
solid and fluid mechanics.
Abstract
This paper introduces a graduate course, continuum mechanics, which is designed for and
taught to graduate students in
a Mechanical Engineering (ME)
program. The significance of continuum mechanics in engineering
education is demonstrated and
the course structure is described.
Methods used in teaching this
course such as topics, class organization, and evaluation instruments are explained. Based on
the student learning outcomes
and feedbacks, the course objectives were achieved. This paper
shows that ME program objectives are well supported by this
course.
Keywords: continuum mechanics,
mechanical engineering, firstyear graduate students
As mentioned before, this course is a problem solving course which focuses on a mathematical study of mechanics of the idealized
continuum mediums. As stated by Petroski [5]
and Reddy [6], undergraduate mathematics
plays an important role in continuum mechanics
research and the continuous material behavior
can be precisely described using modern algebra. As indicated from the course description, a
fundamental basis in differential equations, mechanics of materials, and fluid mechanics are
required before taking this course.
As stipulated by the Accreditation Board for
Engineering and Technology (ABET) [7], the
Mechanical Engineering Department has a set
of eleven educational objectives which are to
be satisfied by the curriculum. This course supports each of those objectives while emphasizing the following:
• Fundamentals. An ability to apply knowledge
of mathematics, science, and engineering in
the field of mechanical engineering.
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17
• Problem Solving. An ability to identify, formulate and solve problems in the field of
mechanical engineering.
• Continuing Education. A recognition of the
need for, and an ability to engage in, lifelong learning in the field of mechanical engineering.
• Engineering Practice. An ability to use the
techniques, skills, and modern tools necessary for the practice of mechanical engineering.
Course Syllabus
Credit Hours: 3
3. Methodology
This course has a well-defined schedule
and syllabus so that the instructor will cover
various topics in the class, which is required by
the breadth of the material and subject matter.
Visual and hands-on learning techniques will
be used wherever possible by the instructor of
this course. In many cases, the ME students
who take this course may have different backgrounds in mathematics. Students will be able
to use class time and available office hours to
discuss special topics with instructor. Descriptions of a course syllabus are presented. As
reflected from the syllabus, mathematical methods and fundamental principles and laws play
an important role in this class.
Continuum Mechanics
Course Goals
The goal of this course is to emphasize the formulation of problems in mechanics along with the
fundamental principles that underlie the governing differential equations and boundary conditions.
Prerequisites by Topic
Differential equations; mechanics of materials; fluid mechanics
Text Book
Reading and homework assignments refer to the required textbook:
T. J. Chung, General Continuum Mechanics, 2007, Cambridge University Press, New York,
NY, USA.
Organization
This class is organized into three 50 minutes sessions per week devoted to lecture/discussion
and problem solving. The difficulty of this course is such that a minimum weekly commitment of
8-10 outside study hours will be required.
Quizzes
Daily quizzes will be given, typically over the previous reading assignment or the material covered in the last class. Missed quizzes cannot be made up.
Exams
Three closed-notes, closed-book exams will be given throughout the semester, which include
two midterm exams and one final exam.
Grading System
Homework Assignments
20%
Quizzes
15%
Midterm Exams
2 @ 20%
Final Exam
25%
Total
100%
Final grades will be calculated according to an absolute scale:
≤ 59 F
Figure 1.
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18
60-62 D-
70-72 C-
80-82 B-
90-92 A-
63-67 = D
73-77 = C
83-87 = B
93-97 = A
68-69 D+
78-79 C+
88-89 B+
98-100 A+
Topic/Activity
1. Introduction to continuum mechanics (2 classes)
- What is continuum mechanics?
- Main themes and assumptions of continuum mechanics.
2. Vectors, tensors and essential mathematics (6 classes)
- Vectors and tensors
- Tensor algebra; summation convention; Kronecker delta; permutation symbol; ε-δ identity.
- Indicial notation; tensor product.
- Matrix operation and linear algebra.
- Transformation of Cartesian tensors.
- Principal values (eigenvalue) and principal directions (eigenvector) of second order tensors.
- Tensor fields and tensor calculus; partial differential operator; subscript comma.
- Integral theorems of Gauss and Stokes
3. Stress principles (3 classes)
- Body and surface forces; mass density.
- Cauchy stress principles; Newton’s 2nd and 3rd laws.
- Stress vector; stress tensor; state of stresses.
- Force and moment equilibrium; stress tensor symmetry.
- Stress transformation laws.
4. Principal stresses and principal axes (3 classes)
- Principal stresses and principal stress directions.
- Maximum and minimum stress values.
- Mohr’s circle for stress.
- Plane stress.
- Spherical stress; deviator stress states.
- Octahedral shear stress.
5. Analysis of deformation (3 classes)
- Particles; configurations; deformation and motion.
- Material and spatial coordinates; find velocity and acceleration for a given motion;
Jacobian matrix.
- Lagrangian and Eulerian descriptions.
- Displacement field.
- Material derivative.
- Deformation gradients, Lagrangian and Eulerian finite strain tensors; Cauchy
deformation tensor.
6. Velocity fields and compatibility conditions (3 classes)
- Infinitesimal deformation theory; strain transformation; strain compatibility equations;
normal strain and shear strain; cubical dilatation.
- Calculate stretch ratios.
- Rotation tensors and stretch tensors; polar decomposition.
- Velocity gradient; rate of deformation and vorticity.
7. Fundamental law and equations (5 classes)
- Material derivative of line elements, areas, and volumes; isochoric motion.
- Balance laws; field equation; constitutive equation.
- Material derivative of line, area, and volume integrals.
- Conversation of mass; continuity equation.
- Linear momentum principle; equation of motion and equilibrium equation.
- Piola-Kirchhoff stress tensors; Lagrangian equation of motion.
- Principle of angular momentum.
- Conversation law of energy; energy equation.
Figure 1. (continued)
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- General constitutive equations.
8. Linear elasticity (6 classes)
- Elasticity; Hooke’s law; strain energy.
- Hooke’s law for isotropic media; elastic constants.
- Elastic symmetry; Hooke’s law for anisotropic media.
- Isotropic elastostatics and elastodynamics; principle of superposition.
- Plane elasticity; difference between plane stress and plain strain.
- Linear thermoelasticity.
- Airy stress function and bi-harmonic equation.
9. Linear viscoelasticity (4 classes)
- Elastic response of solids and viscous flow of fluids.
- Linear viscoelastic materials; constitutive equations in linear differential operator form.
- Construction of viscoelastic models; Maxwell model; Kelvin model; generalized model;
3-parameter solid and fluid model.
- Creep and relaxation; stress relaxation function; Kelvin behavior and Maxwell behavior.
- Principle of superposition; hereditary integrals; distortional and dilatational responses.
- Harmonic loadings; complex modulus and complex compliance.
10. Classical fluids (2 classes)
- Viscous stress tensor; Stokesian and Newtonian fluids.
- Basic viscous flow equations; Navier Stokes equation.
- Barotropic fluids, incompressible fluids, and frictionless fluids.
- Steady flow, irrotational flow, and potential flow.
- Bernoulli equation and Kelvin’s theorem.
11. Nonlinear elasticity and review (3 classes)
- Molecular approach to rubber elasticity; rubber stress; resilience.
- Neo-Hookian material; Mooney-Rivlin model; Ogden material.
12. Examinations (2 classes and 2.5 hour final exam)
Figure 1. (continued)
4. Examples
Two examples are presented here to demonstrate the advantages of continuum mechanics in
solving problems in fluid and solid mechanics.
Example 1. Fluid Mechanics Problem
An incompressible Newtonian fluid maintains a steady flow under the action of gravity down an
inclined plane of slope β. If the thickness of the fluid is perpendicular to the plane h and the pressure
on the free surface is p = p0 (a constant), determine the pressure and velocity fields for this flow.
Solution: Assume v1 = v3 = 0, v2 = v2(x2, x3). By the continuity equation for incompressible flow,
vi,i = 0. Hence, v2,2 = 0 and v2 = v2(x3). Thus, the rate of deformation tensor has components D23 =
D32 = ½(∂v2/∂x3) and all others equal to zero.
The Newtonian constitutive equation is given in this case by
s
ij
= − pd ij + 2m* Dij
from which we calculate



− p
0
0



[s ij ] =  0
m*  ∂v 2
−p

∂x3 



 0 m*  ∂v 2

−p



∂x3 

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Because gravity is the only body force,
b = g (sin beˆ2 − cos beˆ3 )
the equations of motion having the steady flow form: σij,j + ρbi = ρvjvi,j
resulting in component equations:
(for i = 1) –p,1 = 0
(for i = 2) –p,2 + μ*(∂2v2/∂x32) + ρgsinβ = 0
(for i = 3) –p,3 – ρgcosβ = 0
Integrating the last of these gives: p = (ρgcosβ)x3 + f(x2)
where f(x2) is an arbitrary function of integration. At the free surface (x3 = h), p = p0, we have: f(x2)
= p0 – ρgcosβ
and thus: p = p0 + (ρgcosβ)(x3 – h), which describes the pressure in the fluid.
Next, by integrating the middle equation above (for i = 2) twice with respect to x3, we obtain:
with a and b constants of integration. But from the boundary conditions: 1) v2 = 0 when x3 = 0,
therefore b = 0; 2) σ23 = 0 when x3 = h, therefore a = (ρgh/μ*)sinβ.
Finally, therefore, from the equation for v2 we have by the substitution of a = (ρgh/μ*)sinβ, we
obtain:
v2 =
r g sin b
( 2h − x3 ) x3
2 m*
Finish example 1.
Example 2. Solid Mechanics Problem
Consider a special stress function having the form
Φ* = B2x1x2 + D4x1x23
Show that this stress function may be adapted to solve for the stresses in an end-loaded cantilever
beam shown in the Fig. 1. Assume the body forces are zero for this problem.
Figure 1. Cantilever beam loaded at the end by force P
Solutions: It is easily verified, by direct substitution, that
are directly computed as
σ11 = 6D4x1x2; σ22 = 0; ∇ 4f
*
= 0. The stress components
σ12 = - B2 – 3D4x22
These stress components are consistent with an end-loaded cantilever beam, and the constants B2
and D4 can be determined by considering the boundary conditions. In order for the top and bottom
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surfaces of the beam to be stress free, σ12 must be zero at x2 = ±c. Using this condition, B2 is
determined in terms of D4 as B2 = -3D4c2. The shear stress is thus given in terms of single constant
B2
σ12 = -B2 + (B2x22)/c2
The concentrated load is modeled as the totality of the shear stress σ12 on the free end of the beam.
Thus, the result of integrating this stress over the free end of the beam at x1 = 0 yields the applied
force P. in equation form
where the minus sign is required due to the sign convention on shear stress. Carrying out the
integration, we have B2 = 3P/4c so that stress components may now be written as:
σ11 = (-3P/2c3)x1x2;
σ22 = 0;
σ12 = (-3P/4c)( 1 – x22/c2)
But for this beam, the plane moment of inertial of the cross section is I = 2c3/3 so that now
σ11 = (-P/I)x1x2;
σ22 = 0; σ12 = (-3P/2I)( c2 – x22)
in agreement with the results of elementary beam bending theory.
Finish example 2.
5. Discussion
This course was taught by the author to the first-year ME graduate students at University of Louisville in Fall of 2006. As an entry-level graduate course, Continuum Mechanics formed a substantial
background for the students and prepared them for further graduate courses, such as Advanced
Engineering Mathematics (required by all PhD students in Engineering School), Experimental Stress
Analysis, Computational Methods in Fluid Flow and Heat Transfer, Mechanics of Biomaterials, and
Advanced Fluid Mechanics.
Six students enrolled in this class and by the end of that semester, favorable outcomes were
received either from the student feedback or from the course evaluation. The evaluation results
(Table 1) reflected that this course is well designed, so that the students learned fundamental knowledge of continuum mechanics and mastered basic mathematic skills in studying the kinematics and
mechanical behavior of the continuous mediums. The average score of the class is 82/100 with two
students receiving an “A”. After taking this course, five of the six students successfully passed the
comprehensive PhD qualifying exam and are working for their Doctoral degrees at the University
of Louisville. In the qualifying exam, eight problems were given based on the eight major areas in
Mechanical Engineering. Three of them are tightly related to the materials taught in the Continuum
Mechanics, which are Solid Mechanics, Fluid Mechanics, and Advanced Mathematics.
Table 1. Selected course evaluation reports
Query
Strongly
agree
Agree
Undecided
Disagree
Strongly
disagree
I learned a lot in this course
5 (84%)
1 (16%)
0
0
0
This course challenged me to think
Summary
4 (67%)
2 (33%)
0
0
0
Excellent
Good
Average
Fair
Poor
Overall, I would rate this course as
5 (84%)
1 (16%)
0
0
0
Overall, I would rate the effective-ness of
this instructor as
5 (84%)
1 (16%)
0
0
0
Student feedback was also exciting; it showed that the students had learned substantial principles from this class and gained a new appreciation for the study of mechanics. This appreciation
aroused their interests in continuing to work for a Doctoral degree in the ME department. Selected
comments from the students include:
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“Amount of knowledge gained was tremendous.
This class will be very helpful in my career.”
“The class was well organized as always. Dr.
Liu presents a caring, considerate and open attitude toward the students.”
“This is one of the toughest classes, but Dr. Liu
is extremely effective at communicating difficult
concepts in the classroom.”
“I have learned a lot from this class and the instructor has earned my respect!”
With the success of offering the Continuum
Mechanics in University of Louisville, the same
class can also be implemented into ME curriculum as a core graduate course at UL Lafayette
with similar course structure. Due to its low
class/laboratory requirements, no extra equipments and facilities are needed for this class.
Also, the class organization is so flexible that it
can be folded into a short class offered for the
summer semester.
6. Conclusion
This paper describes the design of the
course Continuum Mechanics for the ME graduate students. Its significance and high impact
on engineering education and research have
been fully demonstrated. The course structure
is presented in the paper, from which it can be
found that this course especially focuses on the
understanding and using of advanced mathematics in the study of mechanics within continuous mediums, which makes it unique from other
mechanics classes. As verified in the paper,
this course closely aligns with the ME program,
and major ABET program objectives are satisfied with the establishment of this course.
This course had been taught by the author
in the ME department at University of Louisville and inspiring feedback was received from
the students. Table 1 shows that 84% of students “strongly agreed” that they had learned
a lot from this course, 67% students “strongly
agreed” that this course had challenged them
to think, and 84% students rated this course as
“excellent.” Consequently, the overall teaching
evaluation score for this course was 1.23, which
was markedly above the department average,
1.89, and the university average, 1.75.
From teaching this course, it was also found
that if this course was provided for the first-year
graduate students, it forms a solid background
for further ME graduate courses and makes students competitive for their PhD programs. With
its favorable effects in engineering graduate
program, the author plans to introduce a similar
course at UL Lafayette which will enrich course
offerings in the ME department and the Engi-
neering college, thereby greatly enhancing the
Engineering program in UL Lafayette.
References
1. Mase, G. T., andMase, G. E.(1999). Continuum Mechanics for Engineers, 2nd
Ed.,Boca Raton: CRC Press.
2. Gollub, J. (2003). “Continuum Mechanics
in Physics Education.” Physics Today, December 2003, 10-11.
3. Lagoudas, D. C., Whitcomb, J. D., Miller,
D. A., Lagoudas, M. Z., Shryock, K. J.
(2000).“Continuum mechanics in a restructured engineering undergraduate curriculum”, International Journal of Engineering
Education, 16(4). pp. 301-314.
4. Lagoudas, D. C., Whitcomb, J. D., Miller,
D. A., Lagoudas, M. Z. (1997). “Computer
aided teaching of continuum mechanics to
undergraduates”, Proceedings of ASEE
Gulf-Southwest Annual Conference, The
University of Houston, March 23-25.
5. Petroski, H. J. (1973). “Mathematics in continuum mechanics”, International Journal
of Mathematical Education in Science and
Technology, 4(3). pp. 227-231.
6. Reddy, J. N. (2008). An Introduction to
Continuum Mechanics, New York: Cambridge University Press.
7. “2009-2010 Criteria for Accrediting Engineering Programs.” Accreditation Board
for Engineering and Technology (ABET),
12/1/2008.
Dr. Liu received his PhD degree from the University of Louisville in 2005. He is currently an Assistant
Professor of Mechanical Engineering Department at
University of Louisiana at Lafayette. Dr. Liu’s teaching interests focus on mechanics and kinematics,
statics and dynamics, CAD and FEA, mechanical
and machine design, thermodynamics, etc. To date,
Dr. Liu has published three books, 40 journal articles,
and 12 conference proceedings. Dr. Liu is an editorial board member of International Journal of Vehicle
Structures & Systems, and has served as reviewer
for 17 referred journals and four conferences. Dr. Liu
is also a Professional Engineer registered in Ohio
and holds membership in ASEE, ASME and SAE.
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